Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}5x+9y &= 2 \\ -6x-9y &= -8\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Add the top and bottom equations. $-x = -6$ Divide both sides by $-1$ and reduce as necessary. $x = 6$ Substitute $6$ for $x$ in the top equation. $5( 6)+9y = 2$ $30+9y = 2$ $9y = -28$ $y = -\dfrac{28}{9}$ The solution is $\enspace x = 6, \enspace y = -\dfrac{28}{9}$.